- #1
Understanding L'Hopitals Problem Approximation
- Thread starter sparkle123
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In summary: Having just looked up the history of his name, I now agree with you. I was under the impression that the English spelling L'Hospital came from the direction translation of Hôpital. Apparently he was named L'Hospital originally, with the silent s, but then the French changed their orthography in the mid 18th century and the "os" became a ô. History defeats me again.Thank you to both! :D
- #2
christoff
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Actually, it is L'Hopital's rule. Note that the limit might only exist if i and N have the same sign, or i is negative and N is positive (so as to guarantee we actually have an indeterminate form). We then have three cases; [itex]i<N, i=N, i>N[/itex].
Assuming the above limit exists (that is we're in the situation I mentioned above) applying l'Hopital's rule gives you[tex]\lim_{k\rightarrow ∞}\frac{1-r^{ki}}{1-r^{kN}}=\lim_{k\rightarrow ∞}\frac{-ir^{ki}}{-Nr^{kN}}=\lim_{k\rightarrow∞}\frac{i}{N}r^{k(i-N)}.[/tex]
What happens with the various cases? Compare to the value of the limit [itex]\lim_{k\rightarrow ∞}\frac{r^{ki}}{r^{kN}}[/itex].
Assuming the above limit exists (that is we're in the situation I mentioned above) applying l'Hopital's rule gives you[tex]\lim_{k\rightarrow ∞}\frac{1-r^{ki}}{1-r^{kN}}=\lim_{k\rightarrow ∞}\frac{-ir^{ki}}{-Nr^{kN}}=\lim_{k\rightarrow∞}\frac{i}{N}r^{k(i-N)}.[/tex]
What happens with the various cases? Compare to the value of the limit [itex]\lim_{k\rightarrow ∞}\frac{r^{ki}}{r^{kN}}[/itex].
- #3
Ray Vickson
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sparkle123 said:
The result is false if |r| < 1 because in that case the LHS is 1 but the RHS could be 0, infinity or 1, depending on the values of i and N. For |r| > 1 it follows easily and has nothing at all to do with l'Hospital (that is not l'Hopital---you need an 's' in it).
- #4
christoff
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Ray Vickson said:
True, true; forgot about |r|. Although Ray, I'm a bit confused as to why you say that this has nothing to do with L'Hopital's rule.
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Ray Vickson
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christoff said:
l'HoSpital's rule is not needed, because all we need is to note that r^m - 1 = r^m * [1 - (1/r^m)] and 1/r^m --> 1 as m --> inf.
- #6
christoff
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Ray Vickson said:
Ah, true. Thanks for the clarification. And I really don't like including the s in L'Hôpital; personally I think dropping the accent and adding an s just makes the pronunciation even more confusing to newcomers. Don't want people to think he was a medical doctor or something
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Ray Vickson
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christoff said:Ah, true. Thanks for the clarification. And I really don't like including the s in L'Hôpital; personally I think dropping the accent and adding an s just makes the pronunciation even more confusing to newcomers. Don't want people to think he was a medical doctor or something
Well his name was ... with an 's', but pronounced with no 's'. I do not feel personally empowered to alter the spelling for the sake of convenience, but that's just me.
- #8
christoff
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Ray Vickson said:
Having just looked up the history of his name, I now agree with you. I was under the impression that the English spelling L'Hospital came from the direction translation of Hôpital. Apparently he was named L'Hospital originally, with the silent s, but then the French changed their orthography in the mid 18th century and the "os" became a ô. History defeats me again.
- #9
sparkle123
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Thank you to both! :D
FAQ: Understanding L'Hopitals Problem Approximation
What is L'Hopitals Problem Approximation?
L'Hopitals Problem Approximation is a mathematical technique used to evaluate indeterminate forms, which are expressions where the limit cannot be determined by direct substitution. It was developed by French mathematician Guillaume de l'Hopital in the 18th century.
When should L'Hopitals Problem Approximation be used?
L'Hopitals Problem Approximation should only be used when evaluating limits of indeterminate forms (such as 0/0 or ∞/∞) after other methods have failed, such as factoring or simplifying the expression. It is not a shortcut and should not be used as a first resort.
What is the process for using L'Hopitals Problem Approximation?
The process for using L'Hopitals Problem Approximation involves taking the derivative of the numerator and denominator of the original expression and then evaluating the limit again. If the new limit is still indeterminate, the process can be repeated until a non-indeterminate form is obtained.
Are there any limitations to L'Hopitals Problem Approximation?
Yes, there are limitations to L'Hopitals Problem Approximation. It can only be used for limits involving real numbers and cannot be applied to limits involving complex numbers. It also cannot be used for limits that involve functions with discontinuities or infinite oscillations.
Are there any alternatives to L'Hopitals Problem Approximation?
Yes, there are alternative methods for evaluating limits of indeterminate forms, such as using Taylor series, substitution, or using the squeeze theorem. It is important to try other methods before resorting to L'Hopitals Problem Approximation as it can sometimes lead to incorrect results.